Abstract, registration & information: His short course will cover some first steps in performing computational sample-based inference in inverse problems where the forward map requires solving a PDE (partial differential equation). We will look at some MCMC (Markov chain Monte Carlo) algorithms for drawing samples that are distributed according to the resulting posterior distribution, in few and many dimensions, from simple to state of the art. We will also introduce some basic PDE solvers, and discuss the important finite-rank property of the associated forward map. In the last lecture we will discuss mid-level and high-level models, that are the future of this field. The accompanying practical computer sessions allows participants to get hands-on experience with all these topics.

Abstract, registration & information: Computational modeling has become more and more important in the life sciences. One important class of models are compartment models and ordinary differential equations are widely used to describe the time evolution of the model components in a deterministic way. This compact course will introduce stochastic compartmental modeling. The objectives of the course are to learn a) methods that allow to simulate stochastic models and b) which effects so called intrinsic stochasticity can have on systems dynamics. These effects will make it evident that specific calibration techniques are needed in order to be able to cope with stochastic effects and exploit their information. The course will c) give a flavor of how calibration can be performed. Time will also be devoted to let the participants learn d) when stochastic modeling is necessary and beneficial.
This course will consist of lectures as well as practical exercises. Therefore, participants are encouraged to bring laptops (please contact me in case laptop sharing is desired). There is no prior software or programming experience necessary.

Abstract, registration & information: Optimal sequential Bayesian inference, or filtering, for the state of a dynamical system requires solving a partial differential equation. For low-dimensional, smooth systems the finite-volume method is an effective solver that gives estimates that converge to the optimal continuous-time values. We develop this finite-volume filter, and give numerical examples that show that the filter we develop is able to handle multi-modal filtering distributions. For higher-dimensional systems the curse or dimensionality may be overcome by representing density functions by an interpolated tensor train decomposition. We give examples of filtering for continuous-time and discrete-time systems.

Over the past decades the field of visualization has firmly established itself as an important and constantly expanding discipline within computer science. Computer-based visualization seeks to provide interactive graphical data representations, taking advantage of the extraordinary capability of the human brain to process visual information. Advanced visualization methods now play an important role in the exploration, analysis, and presentation of data in many fields such as medicine, biology, geology, or engineering. This development, however, has also lead to the fact that there is now a vast number of often very specialized techniques to visualize different types of data tailored towards specific tasks. Hence, particularly for non-experts, it becomes increasingly difficult to choose appropriate methods that will provide the optimal answers to their questions.

In this talk, I will discuss previous and ongoing research on how we can explore and navigate the space of visualizations itself. By consider the interplay between data, visualization algorithms, their parameters, perception, and cognition as a complex phenomenon that deserves study in its own right, we are making progress in providing goal-oriented interfaces for visual analysis. For instance, we can make the modification of input parameters of visualization algorithms more intuitive by normalizing their perceived effects over the entire value range, and provide visual guidance about their influence. Furthermore, by incorporating additional knowledge into the visualization process, we can infer information about the goals of a user, and develop smarter systems that automatically suggest appropriate visualization techniques. This line of investigation leads us along the path towards a new type of visual data science, where automated data analysis approaches such as deep learning are tightly coupled with interactive visualization techniques to exploit their complementary advantages for knowledge discovery in data-driven science.

This talk focuses on four active branches in scientific visualization research: topological analysis, feature extraction, volume rendering, and solver visualization. We examine various roles of topological analysis in flow fields and beyond, we investigate feature extraction in higher dimensions and higher order, we extend volume rendering beyond direct geometrical optics, and we extend scientific visualization from the traditional analysis of simulated data to analysis of the numerical solvers that produce the data. On the application side, we discuss the utility of the investigated visualization techniques in the natural and life sciences, and indicate possible directions of future research.

Big, complex, and dynamic image data play an increasing role in science and medicine. This poses important and interesting challenges to scientific visualization, since the traditional visual inspection of raw images is no longer a suitable strategy for their effective and efficient interpretation. Instead, mathematical modeling, feature extraction, and machine learning are required to pre-process the data, and to allow the human user to reason about it at a higher level of abstraction. This talk will illustrate these points with several specific examples from diffusion Magnetic Resonance Imaging, as well as image data from ophthalmic epidemiology.

IWR School "A Crash Course in Machine Learning with Applications in Natural- and Life Sciences (ML4Nature)"

Various Speakers

September 23-27, 2019

ECTS-Points: 3

Abstract, registration & information: The IWR School 2019 gives a crash course in machine learning with applications from Natural Sciences and Life Sciences. We target young researchers from Natural Sciences and Life Sciences who want to learn more about machine learning. A background in machine learning is not required. Besides introducing the basic concepts of machine learning, we teach selected topics in more depth, such as deep learning, metric learning, transfer learning, Bayesian inverse problems, and causality. Experts from machine learning, Natural Science and Life Science explain how these machine learning approaches are utilized to solve problems in their respective fields of research.

Abstract, registration & information: The Tools Seminar provides an opportunity for scientists and students to get proficient information about certain tools useful for their study or research and to exchange their experience and knowledge about those tools with colleagues and fellow students. The term "tool" is understood in a broad sense ranging from tools useful when developing software to more general issues like how to give a good presentation. Particularly, the aim of the Tools Seminar is to provide profound information which go beyond the basic concepts that many people might already be familiar with. But each talk will also include at least a short introduction to allow the participants to learn about tools they might not have used before. So, no matter which level of experience you have with the tools presented, you should be able to learn something new in this seminar!

All talks will be in English!

The preliminary schedule and the registration form can be found here. The participation in the seminar is free of charge, but please register using the registration form for organizational reasons if you plan to participate!

Abstract, registration & information: Learning time-series models is useful for many applications, such as simulation and forecasting. In this study, we consider the problem of actively learning time-series models while taking given safety constraints into account. For time-series modeling we employ a Gaussian process with a nonlinear exogenous input structure. The proposed approach generates data appropriate for time series model learning, i.e. input and output trajectories, by dynamically exploring the input space. The approach parameterizes the input trajectory as consecutive trajectory sections, which are determined stepwise given safety requirements and past observations. We analyze the proposed algorithm and evaluate it empirically on a technical application. The results show the effectiveness of our approach in a realistic technical use case.

Afterwards there will be a Meet & Greet with Pizza and Beer (and non-alcoholic beverages)!

Abstract, registration & information: We consider nonconvex and highly nonlinear mathematical programming problems including finite dimensional nonlinear programming problems as well as optimization problems with partial differential equations and control constraints. We present a novel numerical solution method, which is based on a projected gradient/anti-gradient flow for an augmented Lagrangian on the primal/dual variables. We show that under reasonable assumptions, the nonsmooth flow equations possess uniquely determined global solutions, whose limit points (provided that they exist) are critical, i.e., they satisfy a first-order necessary optimality condition. Under additional mild conditions, a critical point cannot be asymptotically stable if it has an emanating feasible curve along which the objective function decreases. This implies that small perturbations will make the flow escape critical points that are maxima or saddle points. If we apply a projected backward Euler method to the flow, we obtain a semismooth algebraic equation, whose solution can be traced for growing step sizes, e.g., by a continuation method with a local (inexact) semismooth Newton method as a corrector, until a singularity is encountered and the homotopy cannot be extended further. Moreover, the projected backward Euler equations admit an interpretation as necessary optimality conditions of a proximal-type regularization of the original problem. The prox-problems have favorable properties, which guarantee that the prox-problems have uniquely determined primal/dual solutions if the Euler step size is sufficiently small and the augmented Lagrangian parameter is sufficiently large. The prox-problems morph into the original problem when taking the step size to infinity, which allows the following active-set-type sequential homotopy method: From the current iterate, compute a projected backward Euler step by applying either local (inexact) semismooth Newton iterations on the step equations or local (inexact) SQP-type (sequential quadratic programming) methods on the prox-problems. If the homotopy cannot be continued much further, take the current result as a starting point for the next projected backward Euler step. If we can drive the step size all the way to infinity, we can transition to fast local convergence. We can interpret this sequential homotopy method as extensions to several well-known but seemingly unrelated optimization methods: A general globalization method for local inexact semismooth Newton methods and local inexact SQP-type methods, a proximal point algorithm for problems with explicit constraints, and an implicit version of the Arrow--Hurwicz gradient method for convex problems dating back to the 1950s extended to nonconvex problems. We close the talk with numerical results for a class of highly nonlinear and badly conditioned control constrained elliptic optimal control problems with a semismooth Newton approach for the regularized subproblems.

Abstract, registration & information: Scientific computing concerns the development of mathematical models and high-performance software able to describe, simulate and learn the behaviour of complex phenomena. Applications can arise from any area of applied sciences (e.g. engineering, physics, biology, chemistry) and typically retain the challenging task of quantifying high-dimensional uncertainty due to known unknowns and unknown unknowns present in the natural system. When this is added to the computational burden of approximating and solving complex mathematical models, the application of standard inference algorithms, e.g. for parameter estimation, prediction or optimization, becomes quickly unfeasible within a reasonable computational budget.

The aim of the workshop is to bring together researchers working in Uncertainty Quantification, Machine Learning and Bayesian Statistics with a particular focus on high- and infinite-dimensional problems from scientific computing, where the sparsity or uncertainty of data requires an integration of inference and learning algorithms with established physical models, such as partial differential equations. Advances in this complex field of research require a concerted effort from many disciplines, which we hope to foster at the workshop.

This workshop is part of the Thematic Semester Uncertainty Quantification, Machine Learning & Bayesian Statistics in Scientific Computing at MAThematics Center Heidelberg (MATCH) in conjunction with the Excellence Cluster STRUCTURES. The financial support from MATCH and from the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences (HGS MathComp) is gratefully acknowledged.

Abstract, registration & information: Recent years have seen intense development of discretization schemes for incompressible flow problems in two directions. On the one hand, pairs of discrete spaces with a commuting diagram property for the divergence operator have been developed, with the result that it is now possible to compute actually divergence free solutions with reasonable effort. In particular for high Reynolds number flow, this property is important, since the control of the divergence by the gradient is too weak. Alternatively, penalization of the divergence has been investigated thoroughly with the same goal of achieving better solutions for high Reynolds numbers. On the other hand, replacements for the pressure Poisson problem have been developed, allowing for much faster projection of approximate solutions into the divergence free subspace, in particular with high performance computing in mind. The study of stability of such flows and of critical modes requires the solution of nonsymmetric variational eigenvalue problems. Critical modes are characterized by eigenvalues with small real part, which again may suffer from spurious divergence. In order to approximate such eigenvalues, many flow problems must be solved iteratively, which brings fast solvers back into the game.

The goal of this workshop is convening top researchers in the fields of flow and eigenvalue problems in order to understand the interplay of the interacting components better and to profit from recent research of groups with different focus. Furthermore, its aim is intensifying cooperation between the members of PIMS and universities in Germany with a clear focus on common interest.

PIMS - Germany Workshop on Modeling and Analysis of PDEs for Biological Applications

Various Speakers

June 24-26, 2019

ECTS-Points: not yet determined

Abstract, registration & information: This mini-workshop will bring together experts in modeling and analysis of organizing principles of multiscale biological systems such as cell assemblies, tissues and populations, and collective dynamics of cells. We will focus on questions arising in systems biology and medicine which are related to emergence, function and control of spatial structures, cell-cell interactions, and inter-individual heterogeneity in biological dynamics. Mechanisms of symmetry breaking and establishment of spatial patterns in gene expression leading to different differentiation programmes are central issues of developmental biology, while the understanding of their perturbation and deregulation leading to abnormal development is important in cancer research. Evolution of large scale spatial patches such as, for example, systems of vegetation patterns observed in drylands, is essential for ecosystems. Dynamics of appearance and disappearance of such patterns have a direct economic impact. Spatio-temporal dynamics arising through a diffusion field is also a central theme in characterizing the collective response of microbial particles.

Pattern formation is also an important topic in materials science. For example, the nature of formation and evolution of nano-scale structures in energy conversion devices such as fuel cells and solar cells is decisive for the quality of the performance of these devices. Though some of these patterns are well characterized, there are other that we are only beginning to understand. Mathematical modeling is a powerful technique to address key questions and paradigms in diverse model systems and to provide quantitative insights into the role of the nonlinear and nonlocal interactions within the systems and with the external fields as well as of the growth and transport processes and their impact on the observed patterns.
Although applied to specific biological, ecological, chemical, medical or physical systems, mathematical models allow for a comparative analysis of design principles in diverse systems. The focus of this proposed conference is to present and analyze models of partial and integro-differential equations applied to problems of spatio-temporal patterning. The goal of the meeting is to bring together specialists in Germany and from PIMS universities working on different aspects of the field, including mathematical modeling and applications, analysis of the underlying equations as well as numerical simulation, in order to exchange ideas, present new techniques, and identify challenging new research directions of common interest. The focus will be in identifying and understanding of mechanisms of pattern formation including formation of travelling waves, stationary and dynamical patterns, the effect of mechanical-chemical forces on patterns, stability and bifurcation theory, mechanisms underlying collective dynamics in cell signalling, and the emergence of singularities. Applications to developmental biology, ecology, cell-signalling, and materials science will be presented and discussed.

The outcome of the workshop will be two-fold. Firstly, various mathematical methods and techniques presented for diverse types of model PDE sytems in biology, will lead to cross-fertilization and will help solving in tackling problems related to different applications. Secondly, this workshop will identify common research interests and establish new research collaborations on specific projects among researchers at PIMS and at Universities in Heidelberg, Munster and Berlin.